Best Proximity Point Results for Quasi Contractions of Perov Type in Non-Normal Cone Metric Space

Azhar Hussain, Mujahid Abbas, Jamshaid Ahmad, Abdullah Eqal Al-Mazrooei


In this paper, we study the notion of Ciric-Perov quasi contraction and Fisher-Perov quasi contraction and prove some best proximity point theorems for such contractions in the frame work of non-normal regular cone metric spaces. We give an example to support our result. Our results extend and generalized many existing results in literature.


Cone metric spaces; Non-normal cones; Best proximity point; Perov contraction; Spectral radius

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M. Abbas, V. Rakoˇcevic and A. Hussain, Proximal cyclic contraction of perov type on regular cone metric space, J. Adv. Math. Stud. 9(1) (2016), 65 – 71.

M. Abbas, A. Hussain and P. Kumam, A coincidence best proximity point problem in G-metric spaces, Abstract and Applied Analysis 2015(2015), 12 pages, DOI: 10.1155/2015/243753.

M. Abbas and G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces, Journal of Mathematical Analysis and Applications 341(1) (2008), 416 – 420, DOI: 10.1016/j.jmaa.2007.09.070.

A. Amini-Harandi, Best proximity points for proximal generalized contractions in metric spaces, Optim. Lett. 7 (2013), 913 – 921, DOI: 10.1007/s11590-012-0470-z.

M. Al-Khaleel, S. Al-Sharifa and M. Khandaqji, Fixed points for contraction mappings in generalized cone metric spaces, Jordan J. Math. Stat. 5(4) (2012), 291 – 307.

S. S. Basha, Best proximity point theorems generalizing the contraction principle, Nonlinear Analysis 74(2011), 5844 – 5850, DOI: 10.1016/

H. Çakallı, A. Sönmez and Ç. Genç, On an equivalence of topological vector space valued cone metric spaces and metric spaces, Applied Mathematics Letters 25 (2012), 429 – 433, DOI: 10.1016/j.aml.2011.09.029.

Lj. B. Ciric, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267 – 273.

M. Cvetkovic and V. Rakocevic, Quasi-contraction of Perov type, Appl. Math. Comput. 235 (2014), 712 – 722, DOI: 10.1016/j.amc.2014.02.065.

M. Cvetkovic and V. Rakocevic, Fisher Quasi-contraction of Perov type, Journal of Nonlinear and Convex Analysis 16(2) (2015), 339 – 352.

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag (1985).

W. S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Analysis 72(5) (2010), 2259 – 2261, DOI: 10.1016/

A. A. Eldered and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006), 1001 – 1006, DOI: 10.1016/j.jmaa.2005.10.081.

L. Gajic and V. Rakocevic, Quasi-contractions on a nonnormal cone metric space, Functional Analysis and Its Applications 46(1) (2012), 62 – 65, DOI: 10.1007/s10688-012-0008-2.

R. H. Haghi, V. Rakocevic, S. Rezapour and N. Shahzad, Best proximity results in regular cone metric spaces, Rend. Circ. Mat. Palermo 60(2011), 323 – 327, DOI: 10.1007/s12215-011-0050-6.

L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), 1468 – 1476, DOI: 10.1016/j.jmaa.2005.03.087.

S. Jankovic, Z. Kadelburg and S. Radenovic, On the cone metric space: A survey, Nonl. Anal. 74(2011), 2591 – 2601, DOI: 10.1016/

D. Ilic and V. Rakocevic, Quasi-contraction on a cone metric space, Appl. Math. Lett. 22(5) (2009), 728 – 731, DOI: 10.1016/j.aml.2008.08.011.

G. Jungck, S. Radenovic, S. Radojevic and V. Rakoˇcevic, Common fixed point theorems for weakly compatible pairs on cone metric spaces, Fixed Point Theory Appl. 2009 2009, 13 pages, DOI: 10.1155/2009/643840.

Z. Kadelburg, S. Radenovic and V. Rakoˇcevic, Remarks on “Quasi-contraction on a cone metric space”, Appl. Math. Lett. 22(11) (2009), 1674 – 1679, DOI: 10.1016/j.aml.2009.06.003.

W. A. Kirk, P. S. Srinavasan and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (2003), 79 – 89.

A. Latif and M. Abbas and A. Husain, Coincidence best proximity point of Fg-weak contractive mappings in partially ordered metric spaces, Journal of Nonlinear Science and Application 9 (2016), 2448 – 2457, DOI: 10.22436/jnsa.009.05.44.

H. Liu and S. Xu, Cone metric spaces with Banach algebras and fixed point theorems of generalized Lipschitz mappings, Fixed Point Theory and Applications 2013 (2013), Article ID 320, DOI: 10.1186/1687-1812-2013-320.

H. Liu and S. Xu, Fixed point theorem of quasi-contractions on cone metric spaces with Banach algebras, Abstract and Applied Analysis 2013 (2013), Article ID 187348, 5 pages, DOI: 10.1155/2013/187348.

C. Mongkolkeha and P. Kumam, Some common best proximity points for proximity commuting mappings, Optim Lett. 7 (2013), 1825 – 1836, DOI: 10.1007/s11590-012-0525-1.

C. Mongkolkeha, Y. J. Cho and P. Kumam, Best proximity points for Geraghty’s proximal contraction mappings, Fixed Point Theo. and Appl. 2013 (2013), 180, DOI: 10.1186/1687-1812-2013-180.

G. Petrusel, Cyclic representations and periodic points, Studia Univ. Babes¸-Bolyai Math. 50 (2005), 107 – 112.

S. Radenovic and B. E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces, Comput. Math. Appl. 57 (2009), 1701 – 1707, DOI: 10.1016/j.camwa.2009.03.058.

Sh. Rezapour and R. Hamlbarani, Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”, J. Math. Anal. Appl. 345 (2008), 719 – 724, DOI: 10.1016/j.jmaa.2008.04.049.



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